A classical theory of large amplitude, adiabatic collective motion, developed and applied previously to test examples, is transcribed into the language of time-dependent Hartree-Fock theory in order to initiate the application of the new method to problems in nuclear physics. The formulation which emerges can be characterized as a generalized cranking theory, in the sense that the cranking operator cannot be chosen arbitrarily, as in the conventional formulation of this method, but is instead fully constrained by the formalism. A procedure for obtaining approximate solutions to the new equations is described and illustrated with several simplified many-body models. It is inferred that traditional cranking calculations can serve as starting points for more realistic models. This paper also includes additional discussion concerning the calculation of collective masses and the problem of local stability of collective motion, not covered in our previous work.
In recent years we have developed a new mathematical treatment of large amplitude collective motion in the adiabatic limit and formulated a successful approximation method, called the generalized valley approximation. In this paper we discuss its application to adiabatic time-dependent Hartree theory, for which the method is ideally suited. We apply the method first to an exactly solvable limiting case (the Suzuki model), for which we have shown in a previous paper that the usual form of adiabatic time-dependent Hartree theory is not general enough to yield the exact solution. We introduce an extended theory that remedies this deficiency. The modified theory has also been applied to monopole models close to the Suzuki model that are not exactly solvable. The algorithm developed for this case is sufficiently general to serve as a prototype for those necessary to study more complex realistic models of collective motion.
A theory of large amplitude collective motion of a many-particle system is presented, which is relevant, for example, to nuclear fission. The theory is a combination of techniques used in many areas of physics and mathematics. The starting point is the application of the time-dependent Schrödinger equation to generate invariant subspaces of the Hamiltonian in the Hartree–Fock approximation. This is a generalization of the group-theoretical device of generating orbits of a group in the construction of reduced representations. It is shown how solutions of the time-dependent Schrödinger equation can be expressed as instantaneous stationary states of a constrained static Hamiltonian. Thus contact is made with the traditional cranking models and constrained Hartree–Fock theories of large amplitude collective motion. The collective motion is quantized using the Hill–Wheeler–Griffin method of generator coordinates in a basis of generalized coherent states. One is thereby able to exploit much of the theory of harmonic oscillator coherent states, which have been so successfully used in the quantum theory of the laser. The resulting Schrödinger equation for the collective dynamics is expressed both in the Bargmann representation and in the more familiar Schrödinger representation. It is shown that solution of the Schrödinger equation in the small amplitude harmonic approximation reproduces the well-known RPA result. A pilot calculation for 28Si shows that application in large amplitude is also feasible.
The starting point for most studies of large amplitude collective motion in nuclear physics, the time-dependent Hartree-Fock equations, can be mapped to a problem in classical Hamiltonian mechanics, which is the form of the problem studied in this work. For a system with N degrees of freedom, collective motion is identified with motion completely confined to a surface in K<N dimensions. Conditions for the existence of such decoupled surfaces are worked out and a procedure for constructing them is formulated. For most practical problems, where such surfaces do not strictly exist, a concept of approximate decoupling is developed and a test of its accuracy is described. Our results generalize the method of maximum global decoupling found in the earlier literature.
A self-consistent method for evaluation of mass parameters is presented in the framework of the adiabatic limit of the time-dependent Hartree-Fock approximation, reduced to a single collective variable. The corresponding collective path is assumed to be given either by solving a constrained Hartree-Fock problem with a given time-even constraining operator
Q, or by scaling a static Hartree-Fock equilibrium solution. In the former case, once the path is given, a method for solving the equation of motion (of the Hamilton type) is provided, which reduces to a double-constrained Hartree-Fock problem with the time-even constraint
Q and with a time-odd constraining operator
P. In the case of the scaling path, an analytical solution of the Hamilton equation is discussed and the adiabatic mass for the particular case of an isoscalar quadrupole
Q20 mode is given. The operator
P, which is uniquely determined from the knowledge of
Q, has the physical meaning of a momentum operator; it satisfies, together with
Q, a weak quantal conjugation relation. Finally, the connection between the two paths is discussed in terms of generalized random-phase approximation sum rules.NUCLEAR STRUCTURE Evaluation of mass parameters within time-dependent Hartree-Fock approximation in adiabatic limit for single collective variable motion. Discussion and comparison of constrained Hartree-Fock and scaling paths. Expression as a doubly constrained Hartree-Fock problem with momentum operator uniquely defined from coordinate operator. Connection with sum rule and Inglis cranking approaches.
A tractable method is presented for the study of the adiabaticity of a given time-dependent Hartree-Fock solution. In terms of the Baranger-V\'en\'eroni approach, this amounts to extracting the two operators ${\ensuremath{\rho}}_{0}$ and $\ensuremath{\chi}$, for which rather simple expressions are derived. These explicit formulas are exploited to illustrate some implications of the adiabatic approximation itself and several related approximations. In particular, the incompatibility of two current hypotheses for the $\ensuremath{\chi}$ operator (locality and absence of particle-particle, hole-hole matrix elements) is pointed out. Finally, the whole discussion is exemplified by the simple case of uniform translational motion.NUCLEAR STRUCTURE Discussion of adiabatic limit of TDHF approximation, derivation of practical method of study adiabaticity of collective motion from TDHF solutions.
An adiabatic approximation to the selfconsistent collective coordinate method is formulated in order to describe large amplitude collective motions in nuclei with pairing correlations on the basis of the time-dependent Hartree-Fock-Bogoliubov equations of motion. The basic equations are presented in a local harmonic form which can be solved in a manner similar to that of the quasiparticle RPA equations. The formalism guarantees the conservation of nucleon number expectation values. An extension to the multi-dimensional case is also discussed.